Unit 1 : Linear Models : Special Situations

What are the corners of the feasible region? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

(0, 0), (0, 72), (28, 65), and (80, 0)

What are the corners of the region bounded by 5x + 4y ≤ 400 which is not part of the feasible region for the following problem? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

(0, 72), (0, 100), and (28, 65) Portion of the region to the right of the feasible region

The objective function to maximize profit for a linear programming model is P(x, y) = 5x + 10y. Which of the following pairs of coordinates yields the same optimal solution?

(2, 9) and (6, 7) (10, 5) and (4, 8) (4, 8) and (6, 7)

Which of the following constraints would be redundant if added to the existing constraints? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

1.4x + 6y ≤ 432 4x + 4y ≤ 400

Replacement of which one of the following constraints with the 1.5x + 6y ≤ 432 constraint change the feasible region into a point. You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

1.5x + 4.32y ≥ 432

Replacement of which of the following constraints with the 1.5x + 6y ≤ 432 constraint changes the feasible region into an infeasible region? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

1.5x + 4y > 432 4x + 6y ≤ 432 1.5x + 4.32y ≥ 432 1.5x + 4y ≥ 432

Which of the following constraints contradicts the constraint x ≥ 20?

10x + 5y ≤ 180

Which of the following pairs of constraints contradict each other?

4x + 5y ≤ 75; x ≥ 20 x ≤ 15; x ≥ 20

The slope of the objective function f(x, y) = 10x + 4y is

5/2

In the following problem, what should the constraint 5x + 4y ≤ 400 be replaced with to have multiple optimal solutions? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

5x + 7.5y ≤ 400 y ≤ −2/3x + 53 1/3

In terms of a linear programming model, which of the following is true about the feasible region?

Any point in the feasible region is a solution to the system of constraints.

Where is one optimal solution located if a linear programming model has a feasible region with multiple optimal solutions?

Anywhere within the segment of the line between the two corners with the optimal solutions

In the following problem, what must be the value of the decision-making variables? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

Both must be positive Both must be greater than or equal to zero

Which of the following is true about a redundant constraint?

It is not easily traceable at the beginning of a problem. Its existence has no bearing on the optimal solution.

Which of the following is true about an unbounded feasible region?

It usually happens with a minimization problem. It may result in no optimal solution. It may result in a solution of infinity.

What are the corners of the region bounded by 1.5x + 6y ≤ 432 which is not part of the feasible region for the following problem? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

Portion of the region to the right of the feasible region

Which of the following is a possible cause for a special situation?

The feasible region is unbounded. The coordinates of more than one vertex yield the same optimal solution. There is a redundant constraint.

What condition must be met in a linear programming model having a feasible region and multiple optimal solutions?

The slope of at least one constraint should be equal to the slope of the objective function. At least the graph of one of the constraint should be parallel to the graph of the objective function.

Which of the following is a reason why more than one point in the feasible region yields the same optimal solution?

The slope of the objective function is the same as the slope of a constraint. The objective function and one of the constraints have the same slope.

Which of the following is a valid objective function?

f(x, y) = 8x + 7y f(x, y) = 15x + 25y f(x, y) = 5x + 9y

When the objective function has the same slope as one of the constraints, the result is

more than one optimal solution

When two or more constraints contradict each other, the result is

no solution to the problem

A constraint that has no effect on the optimal solution is

redundant

In the following problem, what should the constraint 1.5x + 6y ≤ 432 be replaced with to have multiple optimal solutions? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

y ≤ − 2/3x + 72 4x + 6y ≤ 432

The constraint that contradicts 6x + 8y ≤ 300 is

y ≥ 40